Complex repolarization dynamics in ex vivo human ventricles are independent of the restitution properties

Abstract Aims The mechanisms of transition from regular rhythms to ventricular fibrillation (VF) are poorly understood. The concordant to discordant repolarization alternans pathway is extensively studied; however, despite its theoretical centrality, cannot guide ablation. We hypothesize that complex repolarization dynamics, i.e. oscillations in the repolarization phase of action potentials with periods over two of classic alternans, is a marker of electrically unstable substrate, and ablation of these areas has a stabilizing effect and may reduce the risk of VF. To prove the existence of higher-order periodicities in human hearts. Methods and results We performed optical mapping of explanted human hearts obtained from recipients of heart transplantation at the time of surgery. Signals recorded from the right ventricle endocardial surface were processed to detect global and local repolarization dynamics during rapid pacing. A statistically significant global 1:4 peak was seen in three of six hearts. Local (pixel-wise) analysis revealed the spatially heterogeneous distribution of Periods 4, 6, and 8, with the regional presence of periods greater than two in all the hearts. There was no significant correlation between the underlying restitution properties and the period of each pixel. Conclusion We present evidence of complex higher-order periodicities and the co-existence of such regions with stable non-chaotic areas in ex vivo human hearts. We infer that the oscillation of the calcium cycling machinery is the primary mechanism of higher-order dynamics. These higher-order regions may act as niduses of instability and may provide targets for substrate-based ablation of VF.

Iravanian et al.Higher-Order Periodicity in Human Hearts.

Supplement A: Spatial Filtering
We use a variational method to perform spatial filtering to better control and improve the definition of wavefronts.Let be the normalized fluorescent value (in the 0 to 1 (, ) range with 1 corresponding to peak depolarization) of the pixel at the position and ,   and be the desired smoothed value.We define two discrete energy functions.(, ) First, the smoothness function, Next, the data energy function is defined as Our goal is to minimize the total energy, , where is a super-parameter.

𝐸 = 𝑆 + λ𝐷 λ
Because of the way energy functions are defined, the total energy is in quadratic form.
The resulting minimization problem reduces to solving a sparse linear system, which can be done efficiently using the gradient conjugate method.

Supplement B: Global Analysis
The key to global analysis is to remove the effects of wave propagation to distill the dynamics to a few aggregate channels focused on the repolarization phase.Finally, we generate the spectrograms of the top principal components (Figures 1, panels D and F).The frequency is normalized to the frequency of the driving stimulation; therefore, the 1:1 peak corresponds to the principal action potential propagation.We are mainly interested in the sub-harmonics of the 1:1 peak.The 1:2 peak (located at exactly half the driving frequency) is a sign of period-2 alternans.
Similarly, the 1:4 peak is a marker of the period-4 oscillation in the repolarization phase.

Supplement C: Local Analysis
We discuss the combinatorial algorithm to find the optimal periodicity of each beat in a given input sequence.The algorithm is applied to each valid pixel in the input data and outputs a period map, where the dominant periodicity of each pixel is marked.
For each pixel, the input to the algorithm is a sequence of beats separated by the  upstroke times.Let be a distance function that returns a non-negative real value, (, ,) quantifying the difference between beats and .We assume that satisfies the   (, ) axioms of a distance (or metric) function, meaning that , , and (, ) = 0 (, ) = (, ) . In this paper, we define to be the mean squared (, ) + (, ) ≤ (, ) (, ) difference between beats and .In other applications, may be defined as   (, ) .

|𝐴𝑃𝐷(𝑖) − 𝐴𝑃𝐷(𝑗)|
We start the discussion by presenting the combinatorial algorithm to detect period-2 alternans.Our task is to classify each beat in the input sequence as one of two classes and .For example, can be the long APD beats and the short APD ones.A stable     alternating sequence can be written as .For such a sequence, we can  ••• simply assign to the odd beats and to the even beats.However, the input sequence   may glitch (e.g., two adjacent beats are both short APD) such that the odd/even algorithm fails to work.This problem is especially relevant to higher-order periodicity, where glitches and frameshifts are the rules rather than the exception.The combinatorial algorithm is designed to overcome this shortcoming of simple periodicity detection algorithms.
We find the optimal assignment by setting a weighted directed graph in such a way that the shortest path between the starting vertex (beat 1) and the last vertex (beat )  reveals the optimal assignment (Figure S1 We shift the signals to align the upstrokes (Figure 1, panel A is unshifted signals, and B is frame shifted).Because our signals were recorded while pacing the heart at a stable rate, frameshifting is possible.The core of our global processing routines is dimensionality reduction.We use the standard Principal Component Analysis (PCA) method.Each frame-shifted signal cube is flattened into a two-dimensional matrix (one temporal and one spatial dimension) and subjected to the truncated Singular Value Decomposition.The top few (~5-10) principal components capture most of the dynamics (Figures 1, panels C and E show the top two principal components, marked as W1 and W2).
between two adjacent vertices, i.e., , and is a super-parameter.The  = (,  + 1) η second term is for regularization to prevent spurious high-order detection by favoring shorter periods.The optimal solution is found by assigning to each vertex on the  shortest path from 1 to and to the vertices not on the shortest path.  S1.Schematics of the combinatorial algorithm for local analysis.The graph used to detect alternans, showing (red) and (blue) edges (A). →  + 1  →  + 2 An extended graph to detect up to period-4 oscillations.In addition of the links in A, it also has (green) and (purple) edges (B). →  + 3  →  + 4For detection of period-4, we expand the possible classes of assignment to .{, , , } Now, an ideal input sequence is .We can modify the algorithm above by  •